Holder inequality for symmetric operator spaces and trace property of K-cycles

We prove a general version of the Holder inequality for symmetric operator spaces and symmetric functionals (traces) on such spaces, answering some open questions in the literature. We also prove a general version of the result establishing the trace property of the non-commutative integral defined via an arbitrary positive symmetric functional on a symmetric operator space. Our main tool is the so-called uniform Hardy-Littlewood majorization.